Compositing Operators

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• Author: Diego Inácio
• GitHub: github.com/diegoinacio
• Notebook: compositing_operators.ipynb

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Methods for combining and mixing images.

%matplotlib inline
import matplotlib as mpl
import matplotlib.pyplot as plt
import numpy as np
import imageio

from _utils import *

import warnings
warnings.filterwarnings('ignore')

Input images

---

Given the input images $A$ and $B$ (sRGB colorspace).

p.s.: All the input images are normalized between 0-1.

# Input and normalize to [0, 1]
A = imageio.imread('../_data/fruits.png')/255
histogram(A, bins=2**8, interval=[0, 1])

B = imageio.imread('../_data/pimentos.png')/255
histogram(B, bins=2**8, interval=[0, 1])

Alpha compositing operators

---

n1, n2, _ = A.shape
a = np.zeros((n1, n2, 1))
a[50:350, 50:350] = 1

n1, n2, _ = B.shape
b = np.zeros((n1, n2, 1))
b[200:400] = 1

panel(
    np.array([a[...,0], b[...,0]]), [2, 1],
    text_size=24, text_color='lightgreen',
    texts=[r'$\alpha$', r'$\beta$']
)

Over

---

$$ \large O = A\alpha+B(1-\alpha) $$

O = A*a + B*(1 - a)
histogram(O, bins=2**8, interval=[0, 1])

Atop

---

$$ \large O = A\beta+B(1-\alpha) $$

O = A*b + B*(1 - a)
histogram(O, bins=2**8, interval=[0, 1])

Conjoint over

---

$$ \large O = \begin{cases} A &, \text{if } \alpha > \beta \\ A+\frac{B(1-\alpha)}{\beta} &, \text{otherwise} \end{cases} $$

O = A + B*(1 - a)/b
cond = (a > b)[...,0]
O[cond] = A[cond]
histogram(O, bins=2**8, interval=[0, 1])

Disjoint over

---

$$ \large O = \begin{cases} A+B &, \text{if } \alpha+\beta < 1 \\ A+\frac{B(1-\alpha)}{\beta} &, \text{otherwise} \end{cases} $$

O = A + B*(1 - a)/b
cond = ((a + b) < 1)[...,0]
O[cond] = (A + B)[cond]
histogram(O, bins=2**8, interval=[0, 1])

In

---

$$ \large O = A\beta $$

O = A*b
histogram(O, bins=2**8, interval=[0, 1])

Xor

---

$$ \large O = A(1 - \beta) + B(1 - \alpha) $$

O = A*(1 - b) + B*(1 - a)
histogram(O, bins=2**8, interval=[0, 1])

Blend modes

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Due the input images are in sRGB colorspace, it is necessary to linearize them by the $\gamma = 2.2$ before operating and the output must be transformed to sRGB by inverting the $\gamma$.

$$ \large I' = I^{\gamma} \quad ; \quad O' = O^{\frac{1}{\gamma}} $$

# Gamma
g = 2.2

# Gamma corrected
A_, B_ = A**g, B**g

Average

---

$$ \large O' = \frac{A'+B'}{2} $$

O = (A_ + B_)/2
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Color burn

---

$$ \large O' = 1 - \frac{1-B'}{A'} $$

O = 1 - (1 - B_)/A_

# Deal with zero division
O[O < 0] = 0
O[O > 1] = 1

histogram(O**(1/g), bins=2**8, interval=[0, 1])

Difference

---

$$ \large O' = |A'-B'| $$

O = np.absolute(A_ - B_)
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Division

---

$$ \large O' = \frac{A'}{B'} $$

O = A_/B_
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Exclusion

---

$$ \large O' = A'+B'-2A'B' $$

O = A_ + B_ - 2*A_*B_
histogram(O**(1/g), bins=2**8, interval=[0, 1])

From

---

$$ \large O' = B'- A' $$

O = B_ - A_
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Geometric

---

$$ \large O' = \frac{2A'B'}{A'+B'} $$

O = 2*A_*B_/(A_ + B_)
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Hypot

---

$$ \large O' = \sqrt{A'^2+B'^2} $$

O = (A_**2 + B_**2)**0.5
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Max

---

$$ \large O' = \max(A', B') $$

O = np.maximum(A_, B_)
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Min

---

$$ \large O' = \min(A', B') $$

O = np.minimum(A_, B_)
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Minus

---

$$ \large O' = A'- B' $$

O = A_ - B_
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Multiply

---

$$ \large O' = A'B' $$

O = A_*B_
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Plus

---

$$ \large O' = A' + B' $$

O = A_ + B_
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Screen

---

$$ \large O' = A' + B' - A'B' $$

O = A_ + B_ - A_*B_
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Soft light

---

$$ \large O' = B'(2A' + B'(1 - A'B')) $$

O = B_*(2*A_ + B_*(1 - A_*B_))
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Hard light

---

$$ \large O' = \begin{cases} \text{multiply}&, \text{if } A < \frac{1}{2} \\ \text{screen}&, \text{otherwise} \end{cases} $$

O = A_ + B_ - A_*B_
cond = A_ < 0.5
O[cond] = (A_*B_)[cond]
histogram(O**(1/g), bins=2**8, interval=[0, 1])

Overlay

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$$ \large O' = \begin{cases} \text{multiply}&, \text{if } B < \frac{1}{2} \\ \text{screen}&, \text{otherwise} \end{cases} $$

O = A_ + B_ - A_*B_
cond = B_ < 0.5
O[cond] = (A_*B_)[cond]
histogram(O**(1/g), bins=2**8, interval=[0, 1])